Quantum Chaos Framework

A Python framework for studying quantum chaos and information scrambling in many-body quantum systems, with focus on the Sachdev-Ye-Kitaev (SYK) model and driven Hubbard model.

Overview

This framework provides tools for:

• Constructing SYK and Hubbard model Hamiltonians via Jordan-Wigner transformation
• Computing spectral statistics for quantum chaos diagnostics
• Calculating Out-of-Time-Ordered Correlators (OTOC) for scrambling analysis
• Simulating Floquet dynamics and quasi-energy spectra
• Quantum kernel methods for machine learning

Output

Features

Hamiltonians

• SYK Model: Random all-to-all 4-body Majorana interactions
• Driven Hubbard Model: 1D lattice with time-periodic hopping

Chaos Diagnostics

• Level spacing ratio analysis (GOE/GUE/Poisson classification)
• Fermion parity sector projection
• Disorder averaging with error estimation
• Finite-size scaling extrapolation

Quantum Dynamics

• OTOC calculation for scrambling timescales
• Trotter-Suzuki decomposition for time evolution
• Floquet operator and quasi-energy computation

NISQ Simulation

• Depolarizing, amplitude damping, and phase damping channels
• Realistic noise modeling for near-term quantum devices

Installation

git clone https://github.com/yourusername/quantum-chaos-framework.git
cd quantum-chaos-framework
pip install -r requirements.txt

Requirements

• Python >= 3.8
• NumPy >= 1.20
• SciPy >= 1.7
• Matplotlib >= 3.4
• PennyLane >= 0.30
• scikit-learn >= 1.0

Quick Start

from hamiltonians.syk_hamiltonian import SYKHamiltonian, disorder_average_lsr

# Create SYK model
syk = SYKHamiltonian(n_majorana=8, coupling_strength=1.0, seed=42)

# Get chaos diagnostics
print(f"Ground state energy: {syk.eigenvalues[0]:.4f}")
print(f"Level spacing ratio: {syk.get_level_spacing_ratio():.4f}")
print(f"Level spacing (even parity): {syk.get_level_spacing_ratio_parity('even'):.4f}")

# Disorder averaging for statistics
stats = disorder_average_lsr(n_majorana=8, n_realizations=20)
print(f"⟨r⟩ = {stats['mean']:.4f} ± {stats['stderr']:.4f}")

Project Structure

quantum_chaos_framework/
├── hamiltonians/
│   ├── syk_hamiltonian.py      # SYK model with disorder averaging
│   └── hubbard_hamiltonian.py  # Driven Hubbard model with Floquet
├── circuits/
│   ├── otoc_calculator.py      # OTOC computation methods
│   └── trotter_evolution.py    # Quantum circuit simulation
├── noise/
│   ├── noise_channels.py       # Quantum noise models
│   └── nisq_simulator.py       # NISQ device simulation
├── qml/
│   ├── quantum_kernel.py       # Quantum kernel methods
│   └── quantum_classifier.py   # Variational classifiers
├── utils/
│   ├── jordan_wigner.py        # Fermion-to-qubit mapping
│   └── helpers.py              # Utility functions
├── visualization/
│   └── quantum_visualizer.py   # Plotting tools
├── main.py                     # Example script
└── requirements.txt

Physical Background

SYK Model

The Sachdev-Ye-Kitaev model describes N Majorana fermions with random all-to-all 4-body interactions:

$$H_{SYK} = \sum_{i<j<k<l} J_{ijkl} \chi_i \chi_j \chi_k \chi_l$$

Key properties:

• Maximally chaotic (saturates chaos bound λ ≤ 2πT/ℏ)
• Holographic duality to AdS₂ gravity
• Solvable in large-N limit

Chaos Diagnostics

Level spacing ratio for random matrix classification:

• GOE (chaotic): ⟨r⟩ ≈ 0.530
• GUE (complex chaotic): ⟨r⟩ ≈ 0.603
• Poisson (integrable): ⟨r⟩ ≈ 0.386

OTOC

Out-of-Time-Ordered Correlators measure information scrambling:

$$C(t) = \langle W^\dagger(t) V^\dagger W(t) V \rangle$$

Exponential decay indicates quantum chaos with Lyapunov exponent λ.

References

1. Sachdev, S. & Ye, J. (1993). Gapless spin-fluid ground state in a random quantum Heisenberg magnet. Phys. Rev. Lett.
2. Kitaev, A. (2015). A simple model of quantum holography. KITP talks.
3. Maldacena, J. & Stanford, D. (2016). Remarks on the SYK model. Phys. Rev. D.
4. Maldacena, J., Shenker, S. & Stanford, D. (2016). A bound on chaos. JHEP.

License

MIT License - see LICENSE file for details.

Contributing

Contributions welcome! Please submit issues and pull requests.

Author

Quantum Computing Researcher